5.4 – Properties and Applications
of Logarithims
• Three properties of logarithms correspond to
properties of exponents
• 1) loga(xy) = loga(x) + loga(y)
• 2) loga(x/y) = loga(x) – loga(y)
• 3) loga(xr) = r logax
• These properties can be used to expand
particular expressions
• Example. Use the previous properties to
expand the expression as much as humanly
possible.
• log4(64x3y3)
• Example. Yo! Decompose this mess!
e2 p
• ln( 3 )
q
• We can also use the properties to condense
an expression.
• Example. Condense the following expression.
• ln(x2) – ½ ln(y) + ln (2)
• Example. Condense the following expression.
• 3log72 – 2log74
Change of Base
• Recall…
• With our calculators, we can calculate logs;
but only in base 10
• To overcome this issue, we can use what is
known as the change of base
• Logbx = logax/logab OR ln(x)/ln(b)
• Example. Evaluate the following:
• A) log715
• B) log0.217
• C) log1/5625
Applications
• Example. The pH of a solution is defined as
–log[H3O+], where [H3O+] is the
concentration of hydronium ions in
moles/liter.
• A pH less than 7 is said to be acidic. Greater
than 7 is said to be basic
• Example. A carton of orange juice is found to
have a [H3O+] concentration of 1.58 x 10-4
moles/liter. What is the pH?
• How can we use our equation?
• Example. A person measures the pH in their
pool using a basic kit. The person finds the
[H3O+] to be 2.40 x 10-8 moles per liter. It’s said
to be safe if the pH is between 7.2 and 7.6. Is
it safe to swim in their pool?
• Assignment
• Pg. 425
• 5-11 odd, 19-27 odd, 85, 86, 92,
• Solutions
25) 1/5
31) 9 34) -2 36) No Solution
40) 41.96 44) 5.6 49) 10 60) 4 = log5625
70) ex = log211 72) 81 = 34 78) W = 512
83) e3 = 5x